Course co-ordinator(s): Prof Jennie C Hansen (Edinburgh).
The aims of this course are
- To develop discrete probability models for data
- To understand important features of these models
This course provides an introduction to some of the probability models for inference. The main topics covered are:
- Probability models: sample spaces and events; probability measures and their properties; simple probability calculations.
- Conditioning and Independence: conditional probability; `Chain Rule’ for computing probabilities; Bayes’ Theorem; Partition Theorem; independence of events; probability calculations using concepts of conditioning and independence.
- Random variables and their distributions, including the Binomial, Geometric, Poisson, Hypergeometric, and Indicator variables.
- Expectation and standard deviation of a random variable: definitions and properties of expectation and standard deviation.
1. Introduction to discrete probability models
- Models for random experiments:
- Sample spaces: events, set theory, Venn diagrams, de Morgan’s laws
- Probability functions and the axioms of probability: Choice of probability function, consequences of the axioms.
- Conditional probability and independence of events:
- Definitions of P(A|B) and the definition of independence
- `Chain rule’ for P(A intersection B)
- Bayes Theorem
- Partition Theorem and `tree diagrams’.
2. Special probability models for certain random experiments
- Simple equally likely models
- Sampling without replacement from a finite population:
- ordered samples: model assumptions, calculations and relevant counting arguments
- unordered samples: model assumptions, calculations and relevant counting arguments
- Building models for a sequence of independent sub
- General model assumptions
- Sampling with replacement: calculations and relevant counting arguments
- Bernoulli trials and Binomial models: calculations and relevant counting arguments
- Geometric models
- Other examples
3. Discrete random variables
- Definition of a discrete random variable
- Probability mass functions for discrete random variables
- Expectation and the properties of the expectation of a random variable.
- Variance and the properties of the variance of a random variable.
- Indicator variables
4. Special examples of discrete random variables.
- Binomial and Bernoulli variables
- Geometric and Negative Binomial variables
- Poisson variables
- Hypergeometric variables
- Indicator variables
Learning Outcomes: Subject Mastery
At the end of this course, students should be able to
- Carry out probability calculations for basic discrete probability models using standard techniques including the Partition Theorem, Bayes’ Theorem, and the Chain Rule.
- Work out the distribution, expectation, and standard deviation for a discrete random variable that describes the outcome of a random experiment.
- Identify an appropriate standard random variable to describe the outcome of a random experiment.
- Use the properties of expectation and indicator variables to work out the expected value of a counting variable.
A First Course in Probability by S.M.Ross, 7th ed. (Pearson 2006).
2-hour final exam in May (80%) and continuous assessment (2 assignments) (20%)
SCQF Level: 7.
Help: If you have any problems or questions regarding the course, you are encouraged to contact the lecturer
VISION: further information and course materials are available on VISION